In 1948, Fritz John proposed a theorem stating that every convex body has a unique maximal volume inscribed ellipsoid, known as the John ellipsoid. The John ellipsoid has become fundamental in mathematics, with extensive applications in high-dimensional sampling, linear programming, and machine learning. Designing faster algorithms to compute the John ellipsoid is therefore an important and emerging problem. In [Cohen, Cousins, Lee, Yang COLT 2019], they established an algorithm for approximating the John ellipsoid for a symmetric convex polytope defined by a matrix $A \in \mathbb{R}^{n \times d}$ with a time complexity of $O(nd^2)$. This was later improved to $O(\text{nnz}(A) + d^\omega)$ by [Song, Yang, Yang, Zhou 2022], where $\text{nnz}(A)$ is the number of nonzero entries of $A$ and $\omega$ is the matrix multiplication exponent. Currently $\omega \approx 2.371$ [Alman, Duan, Williams, Xu, Xu, Zhou 2024]. In this work, we present the first quantum algorithm that computes the John ellipsoid utilizing recent advances in quantum algorithms for spectral approximation and leverage score approximation, running in $O(\sqrt{n}d^{1.5} + d^\omega)$ time. In the tall matrix regime, our algorithm achieves quadratic speedup, resulting in a sublinear running time and significantly outperforming the current best classical algorithms.

翻译：1948年，弗里茨·约翰提出定理指出：每个凸体都存在唯一的极大体积内接椭球，即约翰椭球。约翰椭球已成为数学领域的基础概念，在高维采样、线性规划及机器学习中具有广泛应用。因此，设计更快速的约翰椭球计算算法成为一个重要且新兴的研究课题。在[Cohen, Cousins, Lee, Yang COLT 2019]中，研究者建立了针对由矩阵$A \in \mathbb{R}^{n \times d}$定义的对称凸多面体近似约翰椭球的算法，时间复杂度为$O(nd^2)$。该结果随后被[Song, Yang, Yang, Zhou 2022]改进至$O(\text{nnz}(A) + d^\omega)$，其中$\text{nnz}(A)$表示矩阵$A$的非零元数量，$\omega$为矩阵乘法指数。当前最优结果$\omega \approx 2.371$[Alman, Duan, Williams, Xu, Xu, Zhou 2024]。本研究首次提出计算约翰椭球的量子算法，该算法利用谱近似与杠杆值近似量子算法的最新进展，运行时间为$O(\sqrt{n}d^{1.5} + d^\omega)$。在高矩阵维度场景下，本算法实现二次加速，获得亚线性运行时间，显著超越当前最优经典算法。